{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:T7OUOYX5BGYDOET77E54YFGXRO","short_pith_number":"pith:T7OUOYX5","schema_version":"1.0","canonical_sha256":"9fdd4762fd09b037127ff93bcc14d78bb85d33ab94907b894704e252eaa6dca4","source":{"kind":"arxiv","id":"1006.2414","version":3},"attestation_state":"computed","paper":{"title":"The codes and the lattices of Hadamard matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Akihiro Munemasa, Hiroki Tamura","submitted_at":"2010-06-11T22:21:29Z","abstract_excerpt":"It has been observed by Assmus and Key as a result of the complete classification of Hadamard matrices of order 24, that the extremality of the binary code of a Hadamard matrix H of order 24 is equivalent to the extremality of the ternary code of H^T. In this note, we present two proofs of this fact, neither of which depends on the classification. One is a consequence of a more general result on the minimum weight of the dual of the code of a Hadamard matrix. The other relates the lattices obtained from the binary code and from the ternary code. Both proofs are presented in greater generality "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1006.2414","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-11T22:21:29Z","cross_cats_sorted":[],"title_canon_sha256":"7d3284083cd70b899f85195a9c441c814ba43c0ccc076e1cedaef418448c83c2","abstract_canon_sha256":"f66bc3cd75aae8714aef266ef74eaffc956c8177ea59704e4b8f986dcf1e59a8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:31.846075Z","signature_b64":"eZXbS+XI3AYiT60hcnTlCD3BZZm5metlreH2B+MPiaG8orhXcl9iMcZt68gFBPnRXnZzmVhRmR5mkVHYw/4yAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9fdd4762fd09b037127ff93bcc14d78bb85d33ab94907b894704e252eaa6dca4","last_reissued_at":"2026-05-18T00:32:31.845585Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:31.845585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The codes and the lattices of Hadamard matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Akihiro Munemasa, Hiroki Tamura","submitted_at":"2010-06-11T22:21:29Z","abstract_excerpt":"It has been observed by Assmus and Key as a result of the complete classification of Hadamard matrices of order 24, that the extremality of the binary code of a Hadamard matrix H of order 24 is equivalent to the extremality of the ternary code of H^T. In this note, we present two proofs of this fact, neither of which depends on the classification. One is a consequence of a more general result on the minimum weight of the dual of the code of a Hadamard matrix. The other relates the lattices obtained from the binary code and from the ternary code. Both proofs are presented in greater generality "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.2414","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1006.2414","created_at":"2026-05-18T00:32:31.845663+00:00"},{"alias_kind":"arxiv_version","alias_value":"1006.2414v3","created_at":"2026-05-18T00:32:31.845663+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.2414","created_at":"2026-05-18T00:32:31.845663+00:00"},{"alias_kind":"pith_short_12","alias_value":"T7OUOYX5BGYD","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_16","alias_value":"T7OUOYX5BGYDOET7","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_8","alias_value":"T7OUOYX5","created_at":"2026-05-18T12:26:13.927090+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/T7OUOYX5BGYDOET77E54YFGXRO","json":"https://pith.science/pith/T7OUOYX5BGYDOET77E54YFGXRO.json","graph_json":"https://pith.science/api/pith-number/T7OUOYX5BGYDOET77E54YFGXRO/graph.json","events_json":"https://pith.science/api/pith-number/T7OUOYX5BGYDOET77E54YFGXRO/events.json","paper":"https://pith.science/paper/T7OUOYX5"},"agent_actions":{"view_html":"https://pith.science/pith/T7OUOYX5BGYDOET77E54YFGXRO","download_json":"https://pith.science/pith/T7OUOYX5BGYDOET77E54YFGXRO.json","view_paper":"https://pith.science/paper/T7OUOYX5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1006.2414&json=true","fetch_graph":"https://pith.science/api/pith-number/T7OUOYX5BGYDOET77E54YFGXRO/graph.json","fetch_events":"https://pith.science/api/pith-number/T7OUOYX5BGYDOET77E54YFGXRO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T7OUOYX5BGYDOET77E54YFGXRO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T7OUOYX5BGYDOET77E54YFGXRO/action/storage_attestation","attest_author":"https://pith.science/pith/T7OUOYX5BGYDOET77E54YFGXRO/action/author_attestation","sign_citation":"https://pith.science/pith/T7OUOYX5BGYDOET77E54YFGXRO/action/citation_signature","submit_replication":"https://pith.science/pith/T7OUOYX5BGYDOET77E54YFGXRO/action/replication_record"}},"created_at":"2026-05-18T00:32:31.845663+00:00","updated_at":"2026-05-18T00:32:31.845663+00:00"}